reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;

theorem Th9:
  for X being set, F being Field_Subset of X, M being Measure of F,
  A be set st A in F holds (C_Meas M).A <= M.A
proof
  let X be set;
  let F be Field_Subset of X;
  let M be Measure of F;
  let A9 be set;
  assume
A1: A9 in F;
  then reconsider A = A9 as Subset of X;
  reconsider Aseq = (A,{}X) followed_by {}X as Set_Sequence of F by A1,Th8;
A2: Aseq.1 = {}X by FUNCT_7:123;
A3: Aseq.0 = A by FUNCT_7:122;
  A c= union rng Aseq
  proof
    let x be object;
    dom Aseq = NAT by FUNCT_2:def 1;
    then
A4: Aseq.0 in rng Aseq by FUNCT_1:3;
    assume x in A;
    hence x in union rng Aseq by A3,A4,TARSKI:def 4;
  end;
  then reconsider Aseq as Covering of A,F by Def3;
A5: for n being Element of NAT st n <> 0 holds (vol(M,Aseq)).n = 0
  proof
    let n being Element of NAT;
    assume n <> 0;
    then Aseq.n = {} by A2,FUNCT_7:124,NAT_1:25;
    then (vol(M,Aseq)).n = M.{} by Def5;
    hence (vol(M,Aseq)).n = 0 by VALUED_0:def 19;
  end;
  then for n being Element of NAT st 1 <= n holds (vol(M,Aseq)).n = 0;
  then SUM (vol(M,Aseq)) = Ser(vol(M,Aseq)).1 by Th4,SUPINF_2:48
    .= (vol(M,Aseq)).0 by A5,MEASURE7:9;
  then SUM vol(M,Aseq) = M.(Aseq.0) by Def5;
  then
A6: M.A in Svc(M,A) by A3,Def7;
  (C_Meas M).A = inf Svc(M,A) by Def8;
  hence (C_Meas M).A9 <= M.A9 by A6,XXREAL_2:3;
end;
